Aaranav bhai ek help chaiyea
Share
Sign Up to our social questions and Answers Engine to ask questions, answer people’s questions, and connect with other people.
Login to our social questions & Answers Engine to ask questions answer people’s questions & connect with other people.
Answer:
kya ........
Explanation:
First of all, thank you for asking such a beautiful question. I am a fan of Ramanujan.
In Ramanujan’s 2nd Notebook, Chapter XII, Page 108.
Ramanujan, being the genius he was, solve the problem in the most elegant of ways.
I will put it up here for the sake of completeness,
n(n+2)=n1+(n+1)(n+3)−−−−−−−−−−−−−−−√
Let,
f(n)=n(n+2)
f(n)=n1+(n+1)(n+3)−−−−−−−−−−−−−−−√
f(n)=n(1+f(n+1))−−−−−−−−−−−√
f(n)=n1+(n+1)1+(n+2)1+(n+3)−−−−−−−−−√−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√
f(n)=…
That is,
n(n+2)=n1+(n+1)1+(n+2)1+(n+3)1+…−−−−−−√−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√
Putting, n=1 we have,
1+21+31+41+…−−−−−−√−−−−−−−−−−−√−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−√=3
Vijayaraghavan proved that a sufficient condition
for the convergence of the following sequence
a1+a2+…+an−−√−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−√
is that,
limn→∞¯logan2n<∞
ANOTHER FAMOUS METHOD IS, (reverse engineering).
3=9–√=1+8−−−−√=1+2⋅4−−−−−−−√
1+2⋅4−−−−−−−√=1+2⋅16−−√−−−−−−−−−√=1+2⋅1+3⋅5−−−−−−−√−−−−−−−−−−−−−√
Keep going in that way and get the given radical.
Thank you. Cheers! ⌣¨
References: